We call $\triangle{ABC}$ with centroid $G$ [i]balanced[/i] on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B.$ $\triangle{XYZ},$ with $XY=2023$ and $\angle{ZXY}=120^\circ,$ is balanced on $XY.$ What is the maximum value of $XZ$?

Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?

The Professor chooses to assign homework problems from a set of problems labeled $1$ to $100$, inclusive. He will not assign two problems whose numbers share a common factor greater than $1$. If the Professor chooses to assign the maximum number of homework problems possible, how many different combinations of problems can he assign?

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ rows and $ n$ columns , the percentage of $ 1$s on that row or column is not less than $ a$ or greater than $ 100\minus{}a$. Find the largest beautiful number.

In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF \equal{} FG \equal{} GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G); draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C); label("$A$", A, dir(180)); label("$D$", D, dir(180)); label("$B$", B, dir(0)); label("$C$", C, dir(0)); label("$F$", F, dir(270)); label("$E$", E, dir(90)); label("$G$", G, dir(270)); label("$H$", H, dir(140)); label("$J$", J, dir(340)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {5}{2} \qquad \textbf{(B)} \ \frac {35}{12} \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ \frac {7}{2} \qquad \textbf{(E)} \ \frac {35}{8}$

In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$ \[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Let $P$ be a point on side $\overline{AB}$ of equilateral triangle $ABC$. If $BP = 6$ and $CP = 9$, what is the length of $AB$? $\textbf{(A) }2\sqrt5\qquad\textbf{(B) }3+\sqrt6\qquad\textbf{(C) }3\sqrt5\qquad\textbf{(D) }3\sqrt6 + 3\qquad\textbf{(E) }6\sqrt2$

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

Consider an arithmetic progression $a_0,\ldots,a_n$ with $n\ge2$. Prove that $$\sum_{k=0}^n(-1)^k\binom{n}{k}a_k=0.$$

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

Matthew Casertano and Fox Chyatte make a series of bets. In each bet, Matthew sets the stake (the amount he wins or loses) at half his current amount of money. He has an equal chance of winning and losing each bet. If he starts with \$256, find the probability that after 8 bets, he will have at least \$50. [i]Proposed by Jeffrey Tong[/i]

Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths \[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\] (A triangle is non-degenerate if its vertices are not collinear.) [i]Proposed by Bruno Le Floch, France[/i]

Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB,$ and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC.$ Suppose that $BC = 27, CD = 25,$ and $AP = 10.$ If $MP = \tfrac {a}{b}$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.

The fields of an $n\times n$ chess board are colored black and white, such that in every small $2\times 2$-square both colors should be the same number. How many there possibilities are for this?

Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that \begin{align*} cy-bz &\ge 0, \\ az-cx &\ge 0, \\ bx-ay &\ge 0. \end{align*}

What is the probability that a randomly drawn positive factor of $ 60$ is less than $ 7$? $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

If $ a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $ 1988$, show that: $ \sqrt[1988]{\displaystyle\prod_{i,j\equal{}1}^{1988} \left( 1\plus{}\frac{a_i}{a_j} \right)} \ge 2^{1988}$ and determine when equality holds.

Let $a,b,c,d$ be distinct digits such that the product of the $2$-digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$. Find all possible values of $a+b+c+d$.

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

[u]Set 3[/u] [b]3.1[/b] Find the number of distinct values that can be made by inserting parentheses into the expression $$1\,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, -\,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1\,\,\,\,\, - \,\,\,\,\, 1$$ such that you don’t introduce any multiplication. For example, $(1-1)-((1-1)-1-1)$ is a valid way to insert parentheses, but $1 - 1(-1 - 1) - 1 - 1$ is not. [b]3.2[/b] Let $T$ be the answer from the previous problem. Katie rolls T fair 4-sided dice with faces labeled $0-3$. Considering all possible sums of these rolls, there are two sums that have the highest probability of occurring. Find the smaller of these two sums. [b]3.3[/b] Let $T$ be the answer from the previous problem. Amy has a fair coin that she will repeatedly flip until her total number of heads is strictly greater than her total number of tails. Find the probability she will flip the coin exactly T times. (Hint: Finding a general formula in terms of T is hard, try solving some small cases while you wait for $T$.) PS. You should use hide for answers.

Let $z$ be a positive integer that is not divisible by $8$. Furthermore, let $n \geqslant 2$ be a positive integer. Prove that none of the numbers of the form $z^n + z + 1$ is a square number. [i](Walther Janous)[/i]

A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.

(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$. (ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.